Introduction to Geometric Inequalities in General Relativity
课程摘要：This is an introductory topics course for graduate students and young researchers with some background in geometric analysis. We will begin by reviewing the basic mathematical formulation of General Relativity, with the aim of understanding the consequences and proofs (by Schoen/Yau and Witten) of the positive mass theorem. Next we will study a refinement of the positive mass theorem to the case in which spacetime contains black holes. This is known as the Penrose inequality, and provides a lower bound for the mass in terms of the area of the black hole horizons. It has been established in the time symmetric case, but remains open in general. We will study the known proofs of Bray and Huisken/Ilmanen and also the proposals for establishing the general theorem. Other generalizations of the positive mass theorem and Penrose inequality which include charge and angular momentum will be surveyed. The second part of the course will focus on the question of quasi-local mass, which asks for a geometric quantity describing the total mass (gravitation plus matter) content of a bounded region. In particular, the definitions of Bartnik and Wang/Yau will be discussed. If time permits we will study the related issue of the existence problem for black holes, known as the hoop conjecture. The goal is to provide a rigorous mathematical theorem for the heuristic physical intuition that black holes form when too much matter/energy is enclosed in a sufficiently small region. Prof. Marcus Khuri 简介：Dr. Marcus Khuri received his Ph.D from the University of Pennsylvania and has been a professor at Stony Brook University. Dr. Khuri works in the areas of Mathematical Relativity and Geometric Analysis. He has made important contributions to Penrose inequalities and vacuum Einstein equations. He has worked on isometric embedding problems and solved, jointly with Qing Han, a conjecture of Bryant, Griffiths and Yang. Dr. Khuri's recent interests include mixed type Monge-Ampere equations.